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Intertwining operators and complementary series in the class of representations induced from parabolic subgroups of the general linear group over a locally compact division algebra
G. I. Olshanskii^{}
Abstract:
In this paper we study the representations $\operatorname{Ind}(G,P,\pi)$ of the group $G=GL(n,D)$, where $D$ is a locally compact nondiscrete division algebra, that are induced by irreducible representations $\pi$ of an arbitrary parabolic subgroup $P\subset G$. If $D$ is totally disconnected, $\pi$ is assumed to be either supercuspidal (in the sense of HarishChandra; this is the same as absolutely cuspidal in the sense of Jacquet), or onedimensional; we also allow combinations of these cases of a specific sort.
We give a construction of intertwining operators in this class of representations generalizing the construction of Schiffmann, Knapp and Stein. Using these intertwining operators, we prove that for the “principal series” representation $\operatorname{Ind}(G,P,\pi)$ to be contained in the “complementary series” the necessary formal condition of symmetry on $(P,\pi)$ turns out to also be sufficient. If $\pi$ is onedimensional we estimate the width of the “critical interval”. Under certain conditions this estimate is best possible.
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Mathematics of the USSRSbornik, 1974, 22:2, 217–255
Bibliographic databases:
UDC:
519.46
MSC: Primary 22E50, 12A70, 12A80; Secondary 22E45, 12B35 Received: 07.05.1973
Citation:
G. I. Olshanskii, “Intertwining operators and complementary series in the class of representations induced from parabolic subgroups of the general linear group over a locally compact division algebra”, Mat. Sb. (N.S.), 93(135):2 (1974), 218–253; Math. USSRSb., 22:2 (1974), 217–255
Citation in format AMSBIB
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\by G.~I.~Olshanskii
\paper Intertwining operators and complementary series in the class of representations induced from parabolic subgroups of the general linear group over a~locally compact division algebra
\jour Mat. Sb. (N.S.)
\yr 1974
\vol 93(135)
\issue 2
\pages 218253
\mathnet{http://mi.mathnet.ru/msb2970}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=499010}
\zmath{https://zbmath.org/?q=an:0298.22016}
\transl
\jour Math. USSRSb.
\yr 1974
\vol 22
\issue 2
\pages 217255
\crossref{https://doi.org/10.1070/SM1974v022n02ABEH001692}
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This publication is cited in the following articles:

J. H. Bernstein, A. V. Zelevinskii, “Induced representations of the group $GL(n)$ over a $p$adic field”, Funct. Anal. Appl., 10:3 (1976), 225–227

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